Finite Group Databasebeta

← #10844902

Subgroups — #10844902

Order 768.

802 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2568C211yesmaximal, characteristic, complemented
2C2×#5083C211yesmaximal, characteristic
3#82501C211yesmaximal, characteristic
4C2×#5024C211yesmaximal, characteristic, complemented
5#71786C211yesmaximal, characteristic, complemented
6#69683C211yesmaximal, characteristic, complemented
7#69682C211yesmaximal, characteristic, complemented
8#69680C211yesmaximal, characteristic
9#69680C211yesmaximal, characteristic
10#69679C211yesmaximal, characteristic
11#69676C211yesmaximal, characteristic
12#69675C211yesmaximal, characteristic, complemented
13#69674C211yesmaximal, characteristic, complemented
14#69474C211yesmaximal, characteristic
15#69468C211yesmaximal, characteristic, complemented
16#25308—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C3×#400—21no—
23C3×#400—21no—
24C3×#391—21no—
25C3×#391C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×#382—21no—
30C3×C4×C4⋊C4C2211yescharacteristic
31#5231—21no—
32C2×#835C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#835C2211yescharacteristic
38#5083—24no—
39#5024—24no—
40#4993—21no—
41#4989—21no—
42#4989—21no—
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4939C2211yescharacteristic
48#4939C2211yescharacteristic
49#4936C2211yescharacteristic
50#4928C2211yescharacteristic
51#4928C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×#803C2211yescharacteristic
56C2×#803C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#461—31no—
63C2×#446—31no—
64#2580—31no—
65#2568D311yescharacteristic, complemented
66#2068—31no—
67#2066—31no—
68#2055—31no—
69#2055—31no—
70#1973—31no—
71#1968—31no—
72#1960—31no—
73#1949—31no—
74#1935—31no—
75#1933—31no—
76#1762—31no—
77C22×C3×Q8D411yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C22×C3×Q8C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C3×C4⋊Q8—22no—
83C3×C4⋊Q8—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×#215—22no—
98C2×#215—22no—
99C2×#215—22no—
100C2×#215—22no—
101C2×#215—41no—
102C2×#215—41no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C4×D12—22no—
118C4×D12—22no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—21no—
127C3×#95—21no—
128#835—22no—
129#835—22no—
130#835—22no—
131#835—22no—
132#835—41no—
133#835—41no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#809—22no—
139#809—22no—
140#808—41no—
141#807—41no—
142C4⋊(C3⋊C8)—22no—
143C4⋊(C3⋊C8)—22no—
144C4⋊(C3⋊C8)D414yes—
145#803—22no—
146#803—22no—
147C4×C3⋊C8—22no—
148C22×SD16—31no—
149C22×SD16—31no—
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×C4×Q8C2×D311yescharacteristic
152C2×C4×D4—31no—
153#461—64no—
154#446—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169C2×#97—31no—
170C2×C4×C8—31no—
171#400—21no—
172#400—21no—
173#399—61no—
174#393—61no—
175#391—21no—
176#391C2×D311yescharacteristic
177#391C2×D311yescharacteristic
178#391C2×D311yescharacteristic
179#391C2×D311yescharacteristic
180#390—31no—
181#390—31no—
182#389—31no—
183#389—31no—
184#385—61no—
185#385—61no—
186#382—21no—
187#382—31no—
188C4×C4⋊C4C2×D311yescharacteristic
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C23×D3—21no—
194C2×C3×Q8—22no—
195C2×C3×Q8—22no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—41no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2411yescharacteristic
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12#4112yes—
229C2×C4×D3—42no—
230C2×C4×D3—81no—
231C2×C4×D3—81no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—42no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42—42no—
246C3×C42C2×D412yes—
247C3×C42#4112yes—
248#215—44no—
249#215—44no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—81no—
254(C2×D3)⋊C4—81no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261Dic3⋊C4—81no—
262Dic3⋊C4—81no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8C2×D412yes—
273C2×C3⋊C8C2×D412yes—
274C2×C3⋊C8#4112yes—
275C22×Q8#6711yescharacteristic
276C22×Q8C22×D311yescharacteristic
277C22×Q8C22×D311yescharacteristic
278C22×D4—31no—
279C23×C4—61no—
280C2×SD16—62no—
281C2×SD16—62no—
282C2×SD16—62no—
283C2×SD16—62no—
284C2×SD16—121no—
285C2×SD16—121no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—31no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×D4—62no—
295C4×D4—62no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4#6711yescharacteristic
304C2×C4⋊C4#6711yescharacteristic
305C2×C4⋊C4#6711yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C42—21no—
314C2×C42C22×D311yescharacteristic
315C2×C42C22×D311yescharacteristic
316C2×C42C22×D311yescharacteristic
317#107—121no—
318#106—121no—
319C4⋊C8—34no—
320C4⋊C8—62no—
321C4⋊C8—62no—
322#103—62no—
323#103—62no—
324#103—62no—
325#103—62no—
326#103—121no—
327#103—121no—
328#102—62no—
329#102—62no—
330#97—62no—
331#97—62no—
332C4×C8—62no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3C22×D411yescharacteristic
347C22×D3—22no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C3×Q8—44no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×Q8—41no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4#12112yes—
391C2×C3×C4#12112yes—
392C2×C3×C4#13712yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4H(5,𝔽2)11yescharacteristic
398C2×C3×C42−1+411yescharacteristic
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408D12—22no—
409D12—22no—
410D12—41no—
411D12—41no—
412C4×D3—84no—
413C3⋊C8—22no—
414C3⋊C8—22no—
415C3⋊C8—22no—
416C3⋊C8—41no—
417C24—61no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—41no—
433C2×D4—32no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—122no—
447C22×C4—241no—
448C22×C4—241no—
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C2×#6711yescharacteristic
454C22×C4C2×#6711yescharacteristic
455C22×C4C23×D311yescharacteristic
456SD16—124no—
457SD16—124no—
458C2×C8—32no—
459C2×C8—32no—
460C2×C8—32no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—241no—
486C4⋊C4—241no—
487C22⋊C4—122no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C22⋊C4—241no—
491C22⋊C4—241no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—42no—
497C42D3×D412yes—
498C42#23712yes—
499C22×C3C2×#12111yescharacteristic
500C22×C3#54811yescharacteristic
501C22×C3#55011yescharacteristic
502C22×C3C2×#13711yescharacteristic
503C22×C3#57511yescharacteristic
504C22×C3#57711yescharacteristic
505C22×C3#57711yescharacteristic
506C2×D3—42no—
507C2×D3—42no—
508C2×D3—42no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C2×D3—81no—
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—22no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—41no—
519C3×C4—81no—
520C3×C4—81no—
521C3×C4—81no—
522C3×C4—81no—
523C3×C4#52911yescharacteristic
524C3×C4#54211yescharacteristic
525C3×C4#54611yescharacteristic
526C3×C4#54811yescharacteristic
527Dic3—81no—
528Dic3—81no—
529Dic3—81no—
530C23—62no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23C22×#6711yescharacteristic
538Q8—44no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547D4—62no—
548D4—62no—
549D4—121no—
550D4—121no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—244no—
588C2×C4#93912yes—
589C2×C4#93912yes—
590C2×C4#94212yes—
591C2×C4C2×D3×D411yescharacteristic
592C2×C4C2×#23711yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4D3×#4111yescharacteristic
595C2×C4#100911yescharacteristic
596C2×C4#101011yescharacteristic
597C8—62no—
598C8—62no—
599C8—62no—
600C8—121no—
601C2×C3#242611yescharacteristic
602C2×C3#305611yescharacteristic
603C2×C3#305711yescharacteristic
604C2×C3#337211yescharacteristic
605C2×C3#337611yescharacteristic
606C2×C3#340011yescharacteristic
607C2×C3#340111yescharacteristic
608D3—82no—
609C22—122no—
610C22—122no—
611C22—122no—
612C22—241no—
613C22—241no—
614C22—241no—
615C22—241no—
616C22C2×#93911yescharacteristic
617C22C2×#94211yescharacteristic
618C22#582211yescharacteristic
619C22#583111yescharacteristic
620C22#583411yescharacteristic
621C22#583711yescharacteristic
622C22#583711yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—241no—
634C4—241no—
635C4—241no—
636C4#567611yescharacteristic
637C4#567711yescharacteristic
638C4#568011yescharacteristic
639C4#568211yescharacteristic
640C3#2530811yescharacteristic, complemented, Sylow
641C2—242no—
642C2#8273211yescharacteristic
643C2#8299511yescharacteristic
644C2#8339811yescharacteristic
645C2#8366911yescharacteristic
646C2#8367211yescharacteristic
647C2#8368711yescharacteristic
648C2#8369211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2568C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2568D3
hypocenterC3#25308
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2568

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#2568C2semidirect
4. C2×#5024C2semidirect
5. #71786C2semidirect
6. #69683C2semidirect
7. #69682C2semidirect
12. #69675C2semidirect
13. #69674C2semidirect
15. #69468C2semidirect
65. #2568D3semidirect
640. C3#25308semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#25682
C2×#50832
#825012
C2×#50242
#717862
#696832
#696822
#696802
#696792
#696762
#696752
#696742
#694742
#694682
#253083